Multiple-imputation analysis using Stata s mi command

Transcription

1 Multiple-imputation analysis using Stata s mi command Yulia Marchenko Senior Statistician StataCorp LP 2009 UK Stata Users Group Meeting Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

2 Outline 1 Brief overview of multiple imputation 2 Stata 11 s implementation of MI Brief overview of mi Heart-attack data example Creating multiply-imputed data Managing multiply-imputed data Analyzing multiply-imputed data GUI MI control panel Relation to existing user-written commands 3 Summary 4 References Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

3 What is multiple imputation? Multiple imputation (MI) is a simulation-based approach for analyzing incomplete data. MI replaces missing values with multiple sets of simulated values to complete the data, applies standard analyses to each completed dataset, and adjusts the obtained parameter estimates for missing-data uncertainty. The objective of MI is not to predict missing values as close as possible to the true ones but to handle missing data in a way resulting in valid statistical inference (Rubin 1996). Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

4 Why use multiple imputation? It is more flexible than fully-parametric methods, e.g. maximum likelihood, purely Bayesian analysis. It can be more efficient than listwise deletion (complete-cases analysis) and can correct for potential bias. It accounts for missing-data uncertainty and, thus, does not underestimate the variance of estimates unlike single imputation methods. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

5 MI technique The MI technique consists of three steps: 1 Imputation. Replace missing values with M sets of plausible values according to an imputation model (e.g., Rubin 1987; Schafer 1997) to create M completed datasets. 2 Completed-data analysis. Perform primary analysis on each imputed (completed) dataset to obtain a set of completed-data estimates ˆq i and their respective VCEs Û i, i = 1,...,M. 3 Pooling. Consolidate results from the completed-data analyses {ˆq i,û i } M i=1 into one MI inference using Rubin s combination rules (e.g. Rubin 1987, 76). Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

6 Statistical validity of MI MI yields statistically valid inference if an imputation method used in step 1 is proper per Rubin (1987, ). Rubin recommends drawing imputations from a Bayesian posterior predictive distribution of missing data to ensure that imputations are proper. the primary, completed-data analysis used in step 2 is statistically valid in the absence of missing data; see Rubin (1987, ) for details. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

7 Notation and some terminology Original data data containing missing values. With a slight abuse of terminology, by an imputation we mean a copy of the original data in which missing values are imputed. M denotes the number of imputations. m (= 0,..., M) refers to the original or imputed data: m = 0 means original data and m > 0 means imputed data. m = 1 means the first imputation, m = 2 means the second imputation, etc. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

8 Main features Stata 11 s mi command provides full support for all three steps of the MI technique: mi impute performs imputation (step 1); mi estimate performs individual analyses, collects estimates of coefficients and their VCEs, applies Rubin s combination rules to the collected estimates, and reports final results (steps 2 and 3). In addition, mi offers full data management of multiply-imputed data: you can create or drop variables, observations as if you were working with one dataset mi will replicate the changes correctly across the imputed datasets. Other unique features of mi: the ability to store multiply-imputed data in different formats mi data styles; the ability to verify consistency of the data across multiple copies. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

9 mi data system The mi command tracks information about the data which allows full integration of statistical and data-management components. mi records: the format (style) in which MI data will be stored: wide, mlong, flong, or flongsep; the number of imputations, M; the types of registered variables: imputed, passive, regular; information about complete and incomplete observations; see [MI] technical for more detail. mi views system missing values (.) as missing values to be imputed soft missing. All other, extended, missing values are viewed as missing values not to be imputed hard missing values. The mi set and mi register commands are used to set up mi data. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

10 mi data styles To use mi you must declare the storage style. mi supports 4 styles (formats) for storing MI data: flongsep full long and separate imputed data are in separate files, one per imputation; flong full long original and imputed data are in one file, imputations are saved as extra observations; mlong marginal long original and imputed data are in one file, only observations containing imputed values are saved as extra observations. mlong is a memory-efficient version of flong; wide wide original and imputed data are in one file, imputations are saved as extra variables. Some tasks are easier in one style than another. You can switch from one style to another during your mi session by using mi convert. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

11 The role of registered variables in mi mi uses a variable s status to verify its consistency across imputations. You can register variables by using mi register. Registering variables is, in general, not required but highly recommended. mi distinguishes 3 types of variables: imputation (imputed) variables containing soft missings (system missing values), i.e. missing values to be filled in; passive (passive) variables which are functions of imputation and or other passive variables; regular (regular) variables which are the same across imputations; other variables are treated as unregistered. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

12 Always register imputation variables! Important: the status of each observation, complete or incomplete, is determined based on registered imputation variables. An observation in which at least one imputation variable contains a soft missing is marked as incomplete. If no variables are registered as imputed, all observations are treated as complete. You should always register imputation variables. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

13 Example Consider fictional data recording heart attacks. The objective is to examine a relationship between smoking and heart attacks adjusting for age, body mass index, gender, and educational status.. webuse mheart0 (Fictional heart attack data; bmi missing). describe attack smokes age bmi female hsgrad storage display value variable name type format label variable label attack byte %9.0g Outcome (heart attack) smokes byte %9.0g Current smoker age float %9.0g Age, in years bmi float %9.0g Body Mass Index, kg/m^2 female byte %9.0g Gender hsgrad byte %9.0g High school graduate Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

14 We examine data for missing values using misstable.. misstable summarize Obs<. Unique Variable Obs=. Obs>. Obs<. values Min Max bmi Variable bmi contains 22 missing values. We use multiple imputation to perform analysis of the heart-attack data. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

15 Following the MI technique, we need to impute missing values of bmi and then analyze the resulting multiply-imputed data. We will use mi impute and mi estimate to do this but first we need to declare our data to be mi data.. /* check if data are -mi set- */. mi query (data not mi set). /* declare MI data to be stored in the marginal long style */. mi set mlong. /* register bmi as imputation variable */. mi register imputed bmi (22 m=0 obs. now marked as incomplete). /* register other variables as regular */. mi register regular attack smokes age female hsgrad Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

16 We use the regression imputation method to fill in missing values of bmi. We arbitrarily create 5 imputations. We also set the random-number seed for reproducibility.. mi impute regress bmi attack smokes age female hsgrad, add(5) rseed(123) Univariate imputation Imputations = 5 Linear regression added = 5 Imputed: m=1 through m=5 updated = 0 Observations per m Variable complete incomplete imputed total bmi (complete + incomplete = total; imputed is the minimum across m of the number of filled in observations.) Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

17 We perform logistic analysis of the multiply-imputed data using mi estimate: logit.. mi estimate: logit attack smokes age bmi female hsgrad Multiple-imputation estimates Imputations = 5 Logistic regression Number of obs = 154 Average RVI = DF adjustment: Large sample DF: min = avg = max = Model F test: Equal FMI F( 5, ) = 3.39 Within VCE type: OIM Prob > F = attack Coef. Std. Err. t P> t [95% Conf. Interval] smokes age bmi female hsgrad _cons Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

18 Imputation methods Univariate imputation: linear regression for continuous variables mi impute regress predictive mean matching for continuous variables mi impute pmm logistic regression for binary variables mi impute logit ordinal logistic regression for ordinal variables mi impute ologit multinomial logistic regression for nominal variables mi impute mlogit Multivariate imputation: monotone method for multiple variables of different types mi impute monotone multivariate normal regression for multiple continuous variables mi impute mvn Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

19 Imputation methods assumption mi impute assumes that missing data are missing at random; that is, missing values do not carry any extra information about why they are missing than what is already available in the observed data. mi impute creates imputations by simulating from a (approximate) Bayesian posterior predictive distribution of the missing data, following Rubin s recommendation. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

20 mi impute s general syntax mi impute method model spec [, common options method options ] The two main common options are add() and replace. These options allow you to perform the following actions: 1 Create imputations or add new imputations to the existing ones: mi impute..., add(#)... 2 Replace existing imputations with new ones: mi impute..., replace... 3 Replace existing imputations and add new ones: mi impute..., add(#) replace... See [MI] mi impute for more details. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

21 Multivariate imputation: heart-attack data. webuse mheart5s0 (Fictional heart attack data; bmi and age missing). mi describe Style: mlong last mi update 19jun :50:18, 30 days ago Obs.: complete 126 incomplete 28 (M = 0 imputations) total 154 Vars.: imputed: 2; bmi(28) age(12) passive: 0 regular: 4; attack smokes female hsgrad system: 3; _mi_m _mi_id _mi_miss (there are no unregistered variables) Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

22 Data are already set in the mlong style. Two variables, bmi and age, contain missing values and are registered as imputed. Other variables are registered as regular. Data contain no imputations. System variable mi miss records the status of observations (1 means incomplete) based on imputation variables age and bmi. mi m records imputation numbers and mi id records observation identifiers; see [MI] technical for details. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

23 Multivariate imputation: monotone missing pattern. mi misstable nested 1. age(12) -> bmi(28). mi impute monotone (pmm) bmi (regress) age = attack smokes hsgrad female, add(5) Conditional models: age: regress age attack smokes hsgrad female bmi: pmm bmi age attack smokes hsgrad female Multivariate imputation Imputations = 5 Monotone method added = 5 Imputed: m=1 through m=5 updated = 0 age: linear regression bmi: predictive mean matching Observations per m Variable complete incomplete imputed total age bmi (complete + incomplete = total; imputed is the minimum across m of the number of filled in observations.) Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

24 Note that because the data are already mi set, we used mi misstable rather than misstable. We used mi misstable nested to check if variables are nested with respect to missing values. From the output of mi misstable, missing values of age and bmi form a monotone-missing pattern: age is missing only in observations where bmi is missing. bmi does not have any observations with nonmissing values for which age is missing. Therefore, we used mi impute monotone to impute bmi and age. We can also use mi impute mvn to impute bmi and age (as shown on the next slide) but using mi impute monotone is faster because it does not require iteration. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

25 Multivariate imputation: normal regression. mi impute mvn age bmi = attack smokes hsgrad female, replace Performing EM optimization: note: 12 observations omitted from EM estimation because of all imputation variables missing observed log likelihood = at iteration 7 Performing MCMC data augmentation... Multivariate imputation Imputations = 5 Multivariate normal regression added = 0 Imputed: m=1 through m=5 updated = 5 Prior: uniform Iterations = 500 burn-in = 100 between = 100 Observations per m Variable complete incomplete imputed total age bmi (complete + incomplete = total; imputed is the minimum across m of the number of filled in observations.) Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

26 mi impute mvn uses data augmentation, an iterative MCMC method, to impute missing values under a multivariate normal model. mi impute mvn uses estimates from the EM algorithm as starting values for the MCMC procedure. You can supply your own initial values, if needed, using option initmcmc(). The default prior is uniform under which posterior mode estimates and maximum-likelihood estimates are equivalent. You can change the default prior specification using option prior(). The first imputation is drawn after an initial default burn-in period of 100 iterations. You can use option burnin() to choose a different burn-in period. The subsequent imputations are drawn every 100 (the default) iterations apart. You can change the number of iterations between imputations using option burnbetween(). Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

27 Importing existing imputations to mi In the heart-attack example we created imputations using mi impute. What if you need to analyze multiply-imputed data created outside of Stata? 1 Read file(s) containing multiply-imputed data into Stata; see, for example, [D] infile. 2 Use mi import to set up the multiply-imputed data in mi. mi import supports various styles in which multiply-imputed data can be recorded. In particular, mi import nhanes1 imports MI data recorded in the format used by NHANES; see mi import ice imports MI data recorded in the format used by the user-written command ice performing imputation via chained equations. see [MI] mi import for details. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

28 Describing mi data To describe the mi data use mi query to get a short summary of the mi settings; mi describe to get a more detailed report about mi data. mi varying is useful to identify variables that vary over imputations. For example, you can use it to identify imputation and passive variables and then register them using mi register. This command also helps to detect potential problems. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

29 Manipulating mi data Manipulation of mi data can be done in one of two ways: repeating the same data-management command on each imputed dataset; using a data-management routine specialized for multiply-imputed data. For example, specialized routines are needed to append or merge multiply-imputed data. Stata offers both: Use mi xeq:command to perform command on each imputed dataset. Use, e.g. mi append, mi merge, mi reshape to append, merge, and reshape mi data; see [MI] intro (or type help mi) for a list of all mi-specific data-management commands. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

30 Verifying consistency of the mi data mi update mi update is a unique feature of mi. mi update verifies that: the number of observations is consistent in m 0; the number and instances of variables are consistent in m 0; complete/incomplete observations are correctly identified by the imputation variables; regular variables contain the same values in imputed data as in the original data; imputation variables contain the same nonmissing values in imputed data as in the original data; passive variables contain the same values in complete observations in imputed data as in the original data;... ; see [MI] mi update for more detail. mi update is executed automatically each time an mi command is run. It can also be run manually. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

31 Managing observations, variables a quick example Example mi data contain 1 imputation and are saved in the flongsep style. 1. Replace a value:. mi xeq: replace age = 20 in 30 m=0 data: -> replace age = 20 in 30 (1 real change made) m=1 data: -> replace age = 20 in 30 (1 real change made) 2. Drop a variable:. mi xeq: drop female m=0 data: -> drop female m=1 data: -> drop female 3. Alternatively to 1 and 2 above,. replace age = 20 in 30 (1 real change made). drop female. mi update (regular variable female unregistered because not in m=0) (imputed variables updated in 1 obs. in m>0 in order to match m=0 data) Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

32 Managing imputations You can use mi impute to create or add new imputations; mi set m to delete selected imputations; mi add to add imputations from a separate file; mi set M to reset the number of imputations (or create empty imputations in which missing data are not filled in). When performing data manipulation on mi data, remember to use the mi versions of the data-management routines, if they exist; to use mi xeq with routines for which there is no mi prefix; to run mi update periodically to ensure consistency of the mi data. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

33 Estimation using multiple imputation mi estimate is designed to perform analysis of the multiply-imputed data; the data must be mi set and must have at least 2 imputations. Basic syntax: mi estimate: estimation command The above runs estimation command separately on each imputed dataset, and reports the MI estimates of coefficients and their standard errors based on results saved by estimation command. estimation command is one of the supported estimation commands as listed in [MI] estimation. Extended syntax: mi estimate [, options ] : estimation command mi estimate also provides options allowing to select how many and which imputations to use in the computation, to report additional information about the MI estimates, and more. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

34 Other features of mi estimate Other features of mi estimate include the ability to obtain estimates of functions of coefficients; the ability to save individual estimates for later use with mi estimate using without the need of refitting the models on each imputed dataset. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

35 Other features of mi estimate For example, using our earlier heart-attack data example, we save individual estimates to a myest.ster estimation file:. mi estimate, saving(myest): logit attack smokes age bmi female hsgrad We then compute MI estimates of the ratio of coefficients for age and bmi using saved individual estimation results rather than refitting the completed-data models:. mi estimate (ratio: b[age]/ b[bmi]) using myest See [MI] mi estimate and [MI] mi estimate using for more detail. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

36 Analyzing complex multiply-imputed data You can use mi estimate to analyze complex data such as, for example, survey data. In Stata, prior to analyzing complex data, it must be declared. For example, survey data must be svyset, survival data must be stset, and so on. To declare the complex mi data, you should use the corresponding set command with the mi prefix. For example, to declare mi survey data, use. mi svyset... Then, to fit a model on mi survey data, use. mi estimate: svy:... Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

37 Using mi estimate with user-written commands mi estimate provides a way (option cmdok) of applying combination rules to results from estimation commands not officially supported by mi estimate, such as user-written estimation commands.. mi estimate, cmdok : user command It is the user s responsibility to verify that the combination rules are applicable to the results reported by the used command. The user command should also satisfy technical requirements from Writing programs for use with mi in [P] program properties. Authors of user-written commands can also modify their estimation commands to be accepted by mi estimate without specifying cmdok as described in the section mentioned above. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

38 Testing hypotheses After mi estimate, you can test subsets of coefficients, linear or nonlinear hypotheses using mi test and mi testtransform. See [MI] postestimation for examples. mi test and mi testtransform provide the conditional (equal fraction-missing-information, equal FMI) test of Li et al. (1991); the unconditional test of Rubin (1987, 77 78). This test may be preferable when the number of imputations is large and the equal FMI assumption is suspect. small-sample adjustments for the tests as described in Marchenko and Reiter (2009). Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

39 GUI MI control panel The MI control panel, which can be invoked from the Statistics > Multiple imputation menu or by typing. db mi guides you through all the phases of MI. (NEXT SLIDE) Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

40 Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

41 Relation to existing user-written commands User-written commands ice and mim are widely used to perform multiple-imputation analysis in Stata. ice creates imputations using the chained-equation approach of van Buuren et al. (1999). mim analyzes multiply-imputed data. mi estimate and the mi data-management routines cover most estimation and all data-management capabilities of mim. mi does not provide imputation via chained equations and thus ice remains the only implementation of the chained-equation approach. See for more detail. A more detailed FAQ is coming soon. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

42 Summary mi accommodates all steps of the MI technique: mi impute provides univariate and multivariate methods for filling in missing values; mi estimate performs completed-data analysis and combines estimates using Rubin s pooling rules. mi provides full data-management support. mi provides 4 styles for storing MI data and can import from 5 styles. mi verifies consistency of your data at every opportunity. mi offers postestimation features: testing linear or nonlinear hypotheses. mi provides elaborate GUI support MI control panel. mi offers extensive documentation, manual [MI] Multiple imputation. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43

43 References Li, K.-H., T. E. Raghunathan, and D. B. Rubin Large-sample significance levels from multiply imputed data using moment-based statistics and an F reference distribution. Journal of the American Statistical Association 86: Marchenko, Y. V. and J. P. Reiter Improved degrees of freedom for multivariate significance tests obtained from multiply imputed, small-sample data. Stata Journal. Forthcoming. Rubin, D. B Multiple Imputation for Nonresponse in Surveys. New York: Wiley. Rubin, D. B Multiple imputation after 18+ years. Journal of the American Statistical Association 91: Schafer, J. L Analysis of Incomplete Multivariate Data. Boca Raton, FL: Chapman & Hall/CRC. van Buuren, S., H. C. Boshuizen, and D. L. Knook Multiple imputation of missing blood pressure covariates in survival analysis. Yulia Marchenko (StataCorp) Multiple-imputation analysis using mi September 10, / 43